Normal mode analysis Gaucher Disease

Introduction

The protein function often depends on its conformation and dynamical properties. To get a complete picture of the dynamic properties of proteins, the traditional de novo method is known to be Molecular dynamics (MD) simulation. These simulations consider both harmonic and anharmonic motions, and can provide insights into the dynamic of proteins. By taking into account all atoms of a protein, MD simulations shed also light on small motions. However, fine grained MD simulations are computationally very costly which restricts their use to rather short time frames.

An alternative method is Normal mode analysis (NMA) which has become a popular and often used theoretical tool in the study of functional motions in enzymes, viruses, and large protein assemblies<ref name="NMA1">Eric C Dykeman and Otto F Sankey.(2010). Normal mode analysis and applications in biological physics. JOURNAL OF PHYSICS.</ref>. NMA is based on a physical theory about the motion of an oscillation system where all parts within the system move sinusoidally with the same frequency and with a fixed phase relation. By using it to study the protein dynamical motion, the atoms are considered as point masses connected by springs to simulate the inter-atomic forces. NMA uses only harmonic approximations and anharmonic motions are neglected. <ref name="NMA2">Adam D. Schuyler, Gregory S. Chirikjian.(2003). [http://custer.lcsr.jhu.edu/wiki/images/7/76/Schuyler03.pdf Normal mode analysis of proteins: a comparison of rigid cluster modes with Cα coarse graining]. Journal of Molecular Graphics and Modelling.</ref>. Thus, the simulations are not as detailed as MD simulations. Instead, a normal mode refers to the harmonic motion of larger parts of a protein such as domains. Since the normal models can be computed efficiently by matrix decomposition, motions can be simulated over a larger time frame compared to MD simulations. The shape of each normal mode is determined by an eigenvector whose eigenvalue corresponds to the frequency of the normal mode. Normal modes are numbered by their frequency and the lowest frequency mode 7 is often closets to the biological motion of an protein.

In this task, we used different normal mode analysis servers to study the normal modes of chain A of glucocerebrosidase (2nt0).

Technical details are reported in our protocol.

WEBnm@

WEBnm@<ref name="webnma">Siv Hollup, Gisle Salensminde, and Nathalie Reuter. WEBnm@: a web application for normal mode analyses of proteins. BMC Bioinformatics, 6(1):52+, 2005.</ref> is a web server for automatically computing and analysing low-frequency normal modes of proteins. WEBnm@ is based on the Elastic Network Model (ENM) and only takes into account the C-alpha atoms. For computing the 200 lowest frequency mode, WEBnm@ employs the MMTK package[1]. The server offers various methods for investigating the resulting normal modes:

Visualization of the modes
Deformation Energies
Atomic displacements and normalized squared fluctuations
Correlation matrix for investigating the which residue movements are correlated

Deformation Energies

The deformation energy is the potential energy of the motion described by a normal mode. <xr id="tab:defor_enery"/> lists the deformation energy of mode 7 to 20 and <xr id="fig:Eigenvalues_plot"/> shows the respective eigenvalue which corresponds to the frequency of the mode. The deformation energy is positive correlated with the eigenvalue, i.e. modes with a higher frequency also exhibit a higher deformation energy.

</figtable>

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Mode Index	Deformation Energy

7	1663.91
8	2377.48
9	2720.71
10	5191.86
11	5033.67
12	6174.70
13	6360.72
14	6698.31
15	9791.69
16	9534.10
17	10022.74
18	11137.44
19	11592.80
20	12045.03

The deformation energy of mode 7 to 20
of protein structure 2NT0 chain A computed by Webnm@.

Normal mode eigenvalues of 2NT0 chain A computed by Webnma@.

Atomic Displacement

<xr id="fig:atom_disp_7to12"/> shows the normalized square of the displacement of each C-alpha atom for modes 7 to 12. The sum of that value of all the residues is 100. Narrow peaks denote local movements whereas broad peaks indicate larger flexible protein regions.

<xr id="fig:fluc_plot"/> shows the normalized fluctuation of each C-alpha atom calculated by averaging over the atomic displacements of all modes which are weighted by their eigenvalue.

Looking at the displacement plots, in the modes 7, 9, 10 and 11, there are peaks showing throughout the whole protein which suggest that the most parts of this protein are flexible. In mode 8, two individual peaks showing local flexibility at each end of the protein can be found.

</figure>

The Atomic displacements plots for modes 7 to 12 for protein structure 2NT0 chain A from Webnma@.
<figure id="fig:atom_disp_7to12">
<figure id="fig:fluc_plot">

Correlation Matrix

In <xr id="fig:corr_matr"/>, we can see the correlation matrix which presents the correlated movement of the C-alpha atoms of the protein. Each cell shows the isotropic correlation of two residues in the protein on a range from -1 (anti-correlated) via 0 (uncorrelated) to 1 (correlated). Similar to the displacement plots(<xr id="fig:atom_disp_7to12"/>) and fluctuation plot(<xr id="fig:fluc_plot"/>),here we can find again that the red areas around the diagonal indicate that there exist motions throughout the complete protein. And the the region around 0-100 residues and the region around 350-500 residues are somehow relative strongly correlated which reveals a type of isotropic motion of these two regions.

The correlation matrix for protein structure 2NT0 chain A from Webnma@. Each cell in the plot shows the isotropic correlation of two residues in the protein on a range from -1 (anti-correlated) via 0 (uncorrelated) to 1 (correlated).

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Mode Visualization

ElNemo

elNémo is the Web-interface to create the Elastic Network Model which is a fast and simple tool to compute the low frequency normal modes of a protein. It was developed by Yves-Henri Sanejouand and co-workers in 1996.

elNémo allows the user to get the low frequency normal modes for a given protein structure in PDB format. There are different analysis tools available:

compare the collectivity of the modes
view 3-D animations of the protein movement for each mode
identify those residues that have the largest distance fluctuations in a given mode.
Compare B-factors derived from the normal mode decomposition and measured B-factors gives an indication on differences in protein flexibility of the free protein and the protein in a crystallographic environment.

If two conformations of the same protein are uploaded, the user will see the contribution of each mode to the conformational changes (overlap between a protein motion and a normal mode). If two homologous proteins are uploaded, the root mean square distance (RMSD) between all residues and the number of residues that are closer than 3A as a function of mode and perturbation can be computed.

In the following figures, the visualization of model 7 to 11 and their distance fluctuation are shown. The structure views are presented from three different orthologuous viewpoints where the secondary structures are determined from the C-alpha atom positions of the uploaded protein (N-terminal blue, C-terminal red). In the distance fluctuation map, each cell measures the relative movement between a pair of residues. One can find the rigid and the flexible regions of the protein, as well as their relative movements. The regions where the amino acid residues move as rigid bodies are shown in black. And the flexible segments are in blue or red. The blue parts indicate where the distance between two C-alpha atoms increases significantly, and the red parts where the distance decreases.

Normal Mode 7
Normal Mode 7: view through X axis
Normal Mode 7: view through Y axis
The distance fluctuations between all C-alpha atoms.